US senator Ron Wyden of Oregon said: 'Oversight will really be toothless without this kind of information.' Photograph: Joshua Roberts/Reuters

A small group of Democratic legislators fought on the Senate floor on Thursday to shed a sliver of light on the government's domestic surveillance program, seeking to force the National Security Agency to estimate how many US citizens it has spied on.

"I believe that the Senate cannot say that we passed the smell test with respect to vigorous oversight if we don't have some sense of how many Americans … are being swept up under the legislation," Ron Wyden of Oregon, a Democrat, said in debate. "Oversight will really be toothless without this kind of information."

The Senate debate came ahead of a vote on re-authorization of the Foreign Intelligence Surveillance Act (FISA), a 1978 law permitting the government to eavesdrop on correspondence between the United States and foreign addresses.

The House of Representatives has already passed a five-year extension to the law, and senators are likely to renew it by a wide margin.

But critics of the law say it gives the government pretext for eavesdropping on anyone at any time.

Wyden's amendment would require the NSA to put a number on how many Americans have been affected. His Oregon colleague, Jeff Merkley, proposed an amendment that would require the secret court that reviews surveillance requests, known as the FISA court, to inform the public when it makes "important rulings of law".

Intelligence officials have rejected all previous requests for such information, Wyden said.

"The response was: 'It is not reasonably possible to identify the number of people located in the United States whose communications may have been reviewed under the authority of the FISA amendments act,' " he said.

Republican senator Rand Paul, from the libertarian wing of his party, has also proposed an amendment that would require a warrant be issued before government officials could obtain details held by a third party, such as a cloud email provider.

But Dianne Feinstein of California, the outgoing chairwoman of the intelligence committee, argued against the amendments, saying the surveillance programs mostly track foreign nationals.

"No one should think the targets are US persons," Feinstein said. "Thirteen members of the intelligence committee who have voted on this do not believe this is a problem."